Nonexistence of H theorems for the athermal lattice Boltzmann models with polynomial equilibria

We prove that no H theorem exists for the athermal lattice Boltzmann equation with polynomial equilibria satisfying the conservation laws exactly and explicitly. The proof is demonstrated by using the seven-velocity model in a triangular lattice in two dimensions, and can be readily extended to other lattice Boltzmann models in two and three dimensions. Some issues pertinent to the numerical instabilities of the lattice Boltzmann method are disscussed.